Convergence of a Robust Deep FBSDE Method for Stochastic Control

Kristoffer Andersson, Adam Andersson, C. W. Oosterlee

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

In this paper, we propose a deep learning based numerical scheme for strongly coupled forward backward stochastic differential equations (FBSDEs), stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs fails for a simple linear-quadratic control problem, and we motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.

Original languageEnglish
Pages (from-to)A226-A255
Number of pages30
JournalSIAM Journal on Scientific Computing
Volume45
Issue number1
DOIs
Publication statusPublished - Feb 2023

Keywords

  • deep learning
  • FBSDE
  • stochastic control

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